[seqfan] Problem with difference of squares

[Georgi Guninski]

y^2-x^2=n
z^2-x^2=m
n \neq m \neq 0
n,m squarefree

Can the number of solutions in Z be unbounded as n,m vary?

I suppose if the squarefree restriction is dropped the answer is yes via elliptic curves.

Any heuristics?

Probably the question is an open problem.

A possible relaxation might be replacing n,m with c_1 n, c_2 m with small c_1,c_2.

Thanks.

-- 
joro